Joe,
I have not seen any explicit listing of the typical courses of study of those who go into neuroscience. I have been making educated guesses based on the required courses for the graduate programs and upon the specialties of those who have PhDs in neuroscience.
To take the first track, http://neuroscience.berkeley.edu/grad/train/Detailed-Course-Listing.pdf" (PDF). Area A has what amount to advanced undergrad courses in cell biology and genetics. I personally think that one probably doesn't need a degree in biology to understand what is going on here, but a good foundation *is* required. That means University Chemistry II and University Biology II with associated labs at a minimum. If you wanted to get into genetics, you need organic chemistry as well.
Area B is psychology, another subject you are considering. I know far less about this subject area, but the courses listed seem to be survey courses tightly focused on neuroscience, so my impression is that one could know less here and do well. Given that some people come into these programs from the hard sciences, not much background in psychology seems required.
Area C is what I know best. For this reason you should take what I say with grain of salt, because I favor my own subjects! =) Area C has tricky subjects such as information theory, modern statistics, and linear systems. Since this is my own area of expertise, I tend to feel a little critical of the statistical methods of neuroscience practitioners. They probably feel the same way about my limited grasp of the relevant psychology and biology. Good math and programming skills are key in this area.
The second track may be more fruitful. For example,
http://osiris.sunderland.ac.uk/~cs0her/" is an American currently living in the UK who comes at neuroscience from the computer simulation angle. His undergraduate degree was in mathematics.
My experience has been that individuals and institutions are generally open to questions such as yours. If you find someone whose career seems interesting, you might email them and ask about entry into the field. Based on the many kinds of people that work in neuroscience, I think there is a great deal of flexibility in what your undergrad degree is in. If you have a passion for the subject and the ability to distinguish yourself, I think you should do well.